Cambridge touches upon this in two of its definitions of just:
Just for emphasis
We also use just to emphasise an imperative:
- Just shut the door quickly or we’re going to be late. (similar to your 1st example)
Just to soften expressions
We use just in speaking to soften what we say, especially in requests:
- Could you just open the window?
Just is part of quite a few idiomatic expressions,1 one of which is just sayin'. Grammarphobia has an interesting article about it of which I will quote this:
The expression “Just sayin’ ” follows an irritating or annoying or otherwise unpleasant observation. The speaker seems to imply that simply adding “Just sayin’ ” makes everything all right.
Well, it doesn’t.
We briefly referred to this stand-alone expression in a post we wrote a year ago on a similar usage sometimes referred to as a “false front,” “wishwasher,” “but head,” or “lying qualifier.”
Collins has quite a few definitions of just, of which some are labelled as [emphasis].
You use just to indicate that something is no more important, interesting, or difficult, for example, than you say it is, especially when you want to correct a wrong idea that someone may get or has already got.
[emphasis]
You can use just in front of a verb to indicate that the result of something is unfortunate or undesirable and is likely to make the situation worse rather than better.
- Leaving like I did just made it worse.
You use just to emphasize the following word or phrase, in order to express feelings such as annoyance, admiration, or certainty.
[emphasis]
- She just won't relax. I don't see the point in it really. It's just stupid.(annoyance)
- I knew you'd be here. I just knew. (certaninty)
- Isn't he just the most beautiful thing you ever saw? (admiration, affection)
You use just with instructions, polite requests, or statements of intention, to make your request or statement seem less difficult.
[spoken]
- I'm just going to ask you a bit more about your father's business.
(That's the softening Cambridge was speaking about)
And the list goes on. In conclusion, yes, it seems that just can be used as a marker of mood, softener or intensifier.
As For CGEL, as shown in the link provided by @Xanne, it classifies this use of just as a focusing modifier. In the next pages the restrictive adverb only is interpreted grammatically and semantically in detail (mentioning that simply, merely, just are its synonyms). It classifies focus in
- Scopal (They only gave me a SANDWICH for lunch.)
- Informational (Only Kim preferred the ORIGINAL version.)
However, I don't see CGEL dealing with these other layers of meaning present in just, about which you speak about.
1 Such as just about, it's just that..., just like that, let's just say, just as well, just in case etc.
